We consider weak solutions
$u:\Omega _T\to {\open R}^N$
to parabolic systems of the type
$$u_t-{\rm div}\;a(x,t,Du) = 0\quad {\rm in}\;\Omega _T = \Omega \times (0,T),$$
where the function
a(
x,
t,
ξ) satisfies (
p,
q)-growth conditions. We give an a priori estimate for weak solutions in the case of possibly discontinuous coefficients. More precisely, the partial maps
$x\mapsto a(x,t,\xi )$
under consideration may not be continuous, but may only possess a Sobolev-type regularity. In a certain sense, our assumption means that the weak derivatives
$D_xa(\cdot ,\cdot ,\xi )$
are contained in the class
$L^\alpha (0,T;L^\beta (\Omega ))$
, where the integrability exponents
$\alpha ,\beta $
are coupled by
$$\displaystyle{{p(n + 2)-2n} \over {2\alpha }} + \displaystyle{n \over \beta } = 1-\kappa $$
for some
κ ∈ (0,1). For the gap between the two growth exponents we assume
$$2 \les p < q \les p + \displaystyle{{2\kappa } \over {n + 2}}.$$
Under further assumptions on the integrability of the spatial gradient, we prove a result on higher differentiability in space as well as the existence of a weak time derivative
$u_t\in L^{p/(q-1)}_{{\rm loc}}(\Omega _T)$
. We use the corresponding a priori estimate to deduce the existence of solutions of Cauchy–Dirichlet problems with the mentioned higher differentiability property.